Scientific
Interests

Summary:
In my scientific research I study problems in mathematical
analysis and functional analysis, which arise from mathematical
physics. I focus on the spectral analysis of differential and
integral operators with applications to quantum mechanics.

Atoms and
molecules trap electrons at well-defined energy levels only. The
distribution of those eigenvalues has significant impact on the
physical properties of the system. Their study is therefore one of
the main subjects of mathematical physics.

For
sufficiently large structures good approximations for eigenvalues
can be derived in the so-called semi-classical limit. It describes
how quantum mechanical behaviour will transform into the laws of
classical physics. But often such asymptotical calculations extend
to relevant information for the actual quantum regime as well.
Namely, certain spectral characteristics (number or average of
eigenvalues) can be estimated in terms of their classical
counterparts (volume or average of phase spaces).

The
respective results are reflected in the so-called
Cwikel-Lieb-Rosenblum and Lieb-Thirring inequalities. Providing an
intrinsic connection bewteen classical mechanics and "real"
quantum physics, they are of principal importance. They find
numerous applications, for example, in the quantum theory of
many-body systems or in hydrodynamics. As a main line in my
research I have contributed to the theory of these inequalities
[4,5,6,10,13,14,15,18,23,24]. One should point out the results of [13,14],
where we derive optimal constants or give significant improvements
thereof.

In [3,8] we
have studied Schrödinger type operators with periodic
structures. These simulate the behaviour of electrons in crystals.
Given impurities, electrons can form bound states at otherwise
"forbidden" energies. We have shown, that for large
localized perturbations the number of new energy levels does
essentially not depend on the particularities of the structure of
the crystal or external electric or magnetic fields.

In the paper
[9] I study, whether an arbitrary small perturbation of a physical
system will always lead to the trapping of particles (appearance
of virtual bound states). The impact of magnetic fields on this
effect is the subject of [11,12]. Virtual bound states are also
one of the mathematical origins for the trapping effects in
slightly deformed wave-guides. In [9,21] we give a justification for
the so-called edge resonance states in elastic wave-guides.

Ideals of
compact operators form a powerful technical tool to analyse
eigenvalue distributions of partial differential operators. In
[1,2] I have studied certain generalisations of weak
Neuman-Schatten ideals. Applications to Cwikel's theorem can be
found in [4,10,18,22].